Problem: Solve for $x$ : $ 5|x - 10| - 10 = 1|x - 10| + 4 $
Solution: Subtract $ {1|x - 10|} $ from both sides: $ \begin{eqnarray} 5|x - 10| - 10 &=& 1|x - 10| + 4 \\ \\ { - 1|x - 10|} && { - 1|x - 10|} \\ \\ 4|x - 10| - 10 &=& 4 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 4|x - 10| - 10 &=& 4 \\ \\ { + 10} &=& { + 10} \\ \\ 4|x - 10| &=& 14 \end{eqnarray} $ Divide both sides by ${4}$ $ \dfrac{4|x - 10|} {{4}} = \dfrac{14} {{4}} $ Simplify: $ |x - 10| = \dfrac{7}{2}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -\dfrac{7}{2} $ or $ x - 10 = \dfrac{7}{2} $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -\dfrac{7}{2} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -\dfrac{7}{2} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -\dfrac{7}{2} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $2$ $ x = - \dfrac{7}{2} {+ \dfrac{20}{2}} $ $ x = \dfrac{13}{2} $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = \dfrac{7}{2} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& \dfrac{7}{2} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& \dfrac{7}{2} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $2$ $ x = \dfrac{7}{2} {+ \dfrac{20}{2}} $ $ x = \dfrac{27}{2} $ Thus, the correct answer is $x = \dfrac{13}{2} $ or $x = \dfrac{27}{2} $.